The exact connection between the feynman integral and a completely integrable system

Abstract

We review recent attempts to relate the concept of Feynman integral and integrable systems. This constitutes an endeavour on our part in making the Feynman path integral into a mathematically meaningful entity. We then presents a framework which is rooted in the hypothetical relationship between the heuristic concept of Feynman integral in physics and the rigorous mathematical results derived from the theory of (physically significant) completely integrable systems. This idea originates primarily from Wittens (1991) conjecture and Kontsevichs (1992) model which conjecturally able to formulate this remarkable connection. Essentially this link refers to a generator function of intersection numbers on modul space for stable curves (or r-spin curves) and the function of Korteweg-de Vries (or Gelfand-Dikii) hierarchy. In order to display the calculational aspects of this deliberation, certain special models with super potentials are examined.